The following post is by Katharine Downey as a part of George Mason University Math 401, Mathematics Through 3D Printing.
Pentagon Tiling Type 2 – by Katharine Downey
A pentagon tiling is a tiling of a plane in which there are no gaps between any of the pentagons, however all pentagons must be of the same shape.
This is an image of the pentagon tiling I printed for MATH 401, Mathematics Through 3D Printing at George Mason University.
Brief History and Background:
This pattern type is a Type 2 and was found by Reinhardt in 1918. He was the founder of the first five types of pentagonal tilings. There are fifteen types of convex pentagons today that are known to tile a plane monohedrally, the latest of which was found in 2015.
The history of the pentagon tiling discoveries picks up after Rinhardt mush later when Kershner discovered the next three in 1968 who incorrectly stated he had completed the list. Type ten is credited to James in 1975 and shortly after in 1977, a stay-at-home housewife, Marjorie Rice, with a fascination for mathematics found four more. The fourteenth was found some time later in 1985. A fifteenth was found in October of 2015 by Mann, McLoud, and Von Derau from computer algorithm.
Michaël Rao showed that this list of fifteen tilings was complete in that there are no other possible tiling types of pentagon that can tile a plane without any gaps in 2017.
All five types that Reinhardt found can create isohedral tilings, meaning any tile can be mapped to any other tile due to the symmetries. A more formal mathematical explanation of an isohedral tiling is that the automorphism group acts transitively on the tiles.
B. Grünbaum and G. C. Shephard have shown that there are exactly twenty-four distinct types of isohedral tilings of the plane by pentagons according to their own classification scheme. All of these twenty-four types use Reinhardt's tiles, usually with additional conditions necessary for the tiling. There are two tilings by all type two tiles, and nine of the twenty-four tilings are edge-to-edge variations. There are also 2-Isohedral tilings by special cases of type one, type two, and type four tiles, and 3-Isohedral tilings, all edge-to-edge by special cases of type one tiles.
My Tile:
I created my tiling in a program called OpenSCAD, while it is mainly used as an imaging software for 3D printers, it is a very powerful imaging software for geometry and other purposes. After working out the angle and side measurements, I placed a few pentagons in order to see if I calculaded (or coded) the tiling correctly. As some may know from programming/coding various things, a code will hardly ever work the first time which is why I left the angle/side modifiers in. Then I modified the angles or sides in order for the tiling to monohedrally tile the plane.
Here is a piece of the code I used to create my pentagon in OpenSCAD:
This is a piece of the code, not including the angle and side measurements nor the angle/side modifiers. I wrote in order to print out ten type two tiling pentagons.
This is an image of a four-tile primitive unit with one of the tiles out of place to help visualize the shape of the tile.
The angle and side requirements for a type two tile are:
c = e and B + D = 180 degrees
My angle measurements for my tiles are as follows:
Angles: B + D = 180, A = 88, C = 140, E = 132
The wallpaper group symmetry for my printed tiling is pgg (22x), with orbifold notation in parentheses. The wallpaper group for all type two tilings are pgg (22x) unless the mirror image protile tiles are consided distinct, then the symmetry is p2 (2222).
Here are some other Type 2 examples:
The first is pgg (22x) and the second is an edge-to -edge variation with p2 (2222) symmetry.
Sources:
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